What does the Qur’ān and Āhadīth say about the rights/relationship
between husband’s parents and wife and in the same way between wife’s
parents and husband?

Answer:

I am afraid there are no categorical injunctions in the Qur’ān that
specifically mention the rights of the spouse’s parents. The Book of Allah,
however, asserts that you should show kindness to your parents (4:36). What
needs to be appreciated is the fact, that marriage is a contractual
arrangement that binds two persons into one sacred bond. As an obvious
corollary of this agreement, a wife’s relations become her husband’s and a
husband’s relations become hers. It is for this reason that I opine that the
aforementioned directive of treating your parents well extend to your
spouse’s parents as well. You are, indeed, obligated to show kindness to
them as you do to your own. What needs to be kept in mind is the fact that
you and your spouse are standing in a position of ‘give and take’; the time,
care and love that you extend to her family would obviously be returned to
yours.